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Given a full column rank matrix $A \in \mathbb{R}^{n\times m}$. The left pseudo-inverse is $A_{\text{left}}^{-1}=(A^\top A)^{-1}A^{\top}$. Then we have the following relationship $$\|A_{\text{left}}^{-1}\|_2=\frac{1}{\sigma_{\text{min}(A)}}$$.

I have seen this question , and there is also an online problem set that contains this problem. My question is: Where does this problem come from? I want to know whether it is from a certain book so that I can cite it in a proper way.

Thank you very much for your help!!

PT_98
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  • Hint: $\lVert X \rVert_2^2 = \sigma^2_\max(X) = \lambda_\max (X X^T)$. – obareey Sep 04 '23 at 11:28
  • Thanks for the hints, I just wonder if you know where this come from? I need a source of it to cite in my research paper actually – PT_98 Sep 04 '23 at 23:44
  • This is a very standard result. You can find many resources if you search for "singular values" and "(induced) matrix norms". – obareey Sep 06 '23 at 09:32

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